In first place, I would like to say thanks, Alfoncito.
You managed to deliver one of the worst best questions (or one of the best worst questions) in the history of this forum. You ask for polar moment of inertia and then you say its units are kg*m2.
Then, you get several responses from a bunch of guys I know well and who are rarely mistaken.
This confirms my very objective opinion about why civil engineer rules: when you study civil engineering, you have to read through so many subjects that in the end you need to understand, instead of memorizing.
Alfoncito, this I will say once:
The polar moment of inertia
is a quantity you use to estimate resistance to angular torsion
, like in an fixed hinge
, when you want to know when will it break by torsion. It’s a measure of how much resistance to twisting around an axis has a section.
In english: picture yourself grabbing the car (real hard!) by the front bumper and your friends trying to break the car by pushing the rear bumper up. The larger the polar moment, the harder the task of your friends.
The area moment of inertia
or second moment of inertia, is used to estimate resistance to bending
, like in a beam
when you want to know how much will it sag under load. It’s a measure of the resistance of a section of a solid to perpendicular loads.
In english: picture yourself (and your friends) bumping up and down on the roof of the car. The larger the second moment of inertia, the harder is to break the car.
The simple moment of inertia
is used to estimate resistance to rotation
, like in an axle
you want to know how fast rotates, given an energy. It’s analog to inertial mass, but for rotation instead of linear displacements.
For the last time, in english: picture yourself grabbing the car by the front bumper, like Superman, and trying to turn the car in the air around your head. The larger the moment of inertia, the harder is to turn the car around.
Polar moment of inertia: m4
Area moment of inertia: m4
Moment of inertia: Kg*m2
If you weren’t able to distinguish them, my old teacher, Otoniel, would have given you an F in Structures.
So, is the moment of Inertia what you’re asking for, or the polar one?
Anyway, my friend, I believe in the units:
For kg*m2, forget about the worksheets by young engineers or the formulas of the old ones (I share Scotracer fear of the huge formula, the integrals and the such that you'd had to find for a race car, including the density, for the love of Pete!).
So, the simple answer is: Use Autocad MASSPROP function as any sane civil engineer does since his infancy (the icon in the Properties toolbar that displays a solid and a little rule under it).
Draw the car in Autocad and calculate MASSPROP for the XY plane. If you want the POLAR moment, remember that the polar in Z is equal to the second moment in X plus the second moment in Y (if I remember correctly).
Jz = Ix + Iy
If you have to answer it without Autocad (like in an exam or worse), then read the wiki article on moments of inertia, as any sane student does.
Basically, as JTom insinuates, you divide the car or object in small squares and multiply its tiny area (or mass, depending on what you want to calculate) by the distance to the axis of rotation, plane of bending or point of torsion.
BTW, in Autocad, draw the area in the XY plane of the UCS (the funny axis “thingie” in the lower left corner of the screen).
In English: draw the object “flat”.
If you have a 3D drawing or if you want to calculate the number for different axis, rotate your car in the UCS plane accordingly and take a section of it.
Now, I wonder if I made a mistake… I’m sure somebody will notice, but what the heck, he will have to read the damn wiki article… and come back to explain if Otoniel missed me.
Besides, I haven't used the MASSPROP function that much, perhaps somebody knows how to calculate the different moments of inertia from it (the function gives you the POLAR moment of inertia, if I'm not mistaken). I vaguely remember that you divide by the radius of gyration squared (maybe) to get what I've called the simple moment of inertia, but I'm not sure.
You know, I'm civil, I use it to get the properties of a beam under bending loads and the such, I don't rotate the beams that much (but I can distinguis both cases, unlike mechanicals